Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which ${\rm R}^4$ can be decomposed, an analogue of the Cartan-Weyl connection-type form of the Hilbert-Einstein action in the usual continuum GR. In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog $γ_{\rm F}$ of the Barbero-Immirzi parameter $γ$. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate $γ_{\rm F}$.
First order minisuperspace model for the Faddeev formulation of gravity
考虑了由10个向量场或10维四元组构成度规的广义相对论的Faddeev表述。在部分使用的场方程，这个理论的结果在通常的GR。早些时候，我们已经提出了一些minisuperspace模型的Faddeev制定的四元组字段是分段常数的多面体，如4-单纯形，或者说，长方体，其中${\rm I \hspace{-3pt} R}^4$可以分解。现在我们研究这个（离散）理论的某些表示，它是通常连续统GR中希尔伯特-爱因斯坦作用量的Cartan-Weyl连接型形式的类似物。
Some minisuperspace model for the Faddeev formulation of gravity
我们考虑广义相对论的Faddeev公式，其中度规由十个矢量场或一个$4 \times 10$四元组组成。这个公式在部分使用场方程时，简化为通常的广义相对论。Faddeev作用的一个显著特点是它在不连续域上的有限性。这允许引入其minissuperspace公式，其中矢量场在${\rm I \hspace{-3pt} R}^4$上的任何地方都是恒定的，除了测量零点集（分段恒定场）。这些场通过在e中选择的它们的常数值{独立地\it}来参数化。例如，在一个实施例中，4-单形，或者说，平行六面体，其中${\rm I \hspace{-3pt} R}^4$可以分解。我们还考虑了分段常向量场逼近固定光滑向量场的情形。我们检查，如果区域中的矢量场是恒定的任意小，迷你超空间的行动和运动方程往往是连续法捷耶夫的。
Faddeev gravity using a $d$-dimensional tetrad (normally $d = 10$) is classically equivalent to general relativity (GR). The discrete Faddeev gravity on the piecewise flat spacetime normally assumes slowly varying metric and tetrad from vertex to vertex. Meanwhile, Faddeev action is finite (although not unambiguously defined) for discontinuous tetrad fields thus allowing, in particular, to consider a surface as consisting of virtually independent elementary triangles, and its area spectrum as the sum of elementary area spectra. In the discrete connection form, area tensors are canonically conj…
Faddeev variant of embedding theory is an example of using the embedding approach for the description of gravity. In the original form of the embedding approach, the gravity is described by an embedding function of a four-dimensional surface representing our spacetime. In Faddeev variant, the independent variable is a non-square vielbein, which is a derivative of embedding function in embedding theory. We study the possibility of the existence of extra solutions in Faddeev variant, which makes this theory non-equivalent to GR. To separate the degrees of freedom corresponding to extra matter, w…
We show how to systematically apply the Faddeev-Jackiw symplectic method to General Relativity (GR) and to GR extensions. This provides a new coherent frame for Hamiltonian analyses of gravitational theories. The emphasis is on the classical dynamics, uncovering the constraints, the gauge transformations and the number of degrees of freedom; but the method results are also relevant for canonical quantization approaches. We illustrate the method with three applications: GR and to two Brans-Dicke cases (the standard case $ω\not= - 3/2$ and the case with one less degree of freedom, $ω= - 3/2$). W…
The propagators for the Faddeev-Popov (FP) ghosts in Yang-Mills theory and perturbative gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. An IR cutoff in the momentum space to regularize these divergences breaks the de Sitter invariance. These IR divergences are due to the spatially constant modes in the Yang-Mills case and the modes proportional to the Killing vectors in the case of perturbative gravity. It has been proposed that these IR divergences can be removed, with the de Sitter invariance preserved, by first regularizing them with an additional mass ter…
A Lagrangian formulation for Rastall gravity and a covariant formulation for unimodular gravity
Faculdade de Ciências e Tecnologias, Universidade de Cabo Verde（佛得角大学科学与技术学院）；Núcleo Cosmo-ufes & Departamento de Física, Universidade Federal do Espírito Santo（圣埃斯皮里图联邦大学宇宙核与物理系）；Centro de Ciências Naturais e Humanas, Universidade Federal do ABC（ABC联邦大学自然与人文学中心）
We present the Hamiltonian formulation of the Ghost Free Mimetic Massive Gravity theory. The linearized theory is studied and the Hamiltonian equations of motion are analyzed. Poisson brackets are computed and closure is proved. To prove that this theory is ghost-free, the number of degrees of freedom is analyzed showing that we have only five degrees of freedom.
We analyze the gravitational waves propagation in metric $f(R)$ theories of gravity. In particular, adopting a gauge invariant formalism we clearly determine the exact propagating degrees of freedom. Then, investigating their effects on test masses via geodesic deviation equation, we show that the additional dynamical degree contained in such extended formulations is actually responsible for two distinguished polarizations, corresponding to a breathing and a longitudinal mode, respectively.
There are at least two ways to encode gravity into geometry: Einstein's general theory of relativity (GR) for the metric tensor, and teleparallel gravity, where torsion as opposed to curvature encodes the dynamics of the gravitational degrees of freedom. The main purpose of the paper is to explore the relation between loop gravity and teleparallel gravity. We argue that these two formulations of gravity are related to two different discretizations of the Einstein--Cartan action, which were studied recently in the literature. The first discretization leads to the \emph{loop gravity} kinematical…
